For all the arguments against Common Core (I personally love Common Core, but that's a topic for another post), the change in standards has shifted our focus from product to process in a way that was much needed. We don't just do long division anymore; we do repeated subtraction and partial quotients. We don't just do multi-digit multiplication anymore; we do partial products and use the distributive property to break apart the products. And in my classroom, we don't just cross multiply anymore; we multiply each fraction by a whole number.
I agree that students shouldn't just learn a "trick" but cross-multiplication is not a trick. It works because you're taking a shortcut for multiplying each fraction by a whole number fraction with a numerator and denominator that matches the denominator of the opposite fraction. The numbers you write at the top when you "cross-multiply" are simply the numerators of the equivalent fractions that is produced in this process. Once my students understand this, I don't mind if they just write the number at the top and "cross-multiply" the same way I don't mind if they take shortcuts in other math operations.
Today I worked with a student after school on comparing fractions. (That's the actual paper we used to work on.) While we were "cross-multiplying" he said, "No one's ever really explained that to me before." As a teacher, no one had ever explained it to me either; I just took the time to figure out why it works. That's the thing about math-- there aren't really any "tricks." Everything works for a reason. Our job is to make sure they understand all those reasons, one by one.
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